Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function \(G_s(z_1,z_2)\) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable \(z_1\) over all CM points of a fixed discriminant \(d_1\) (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant \(d_2\). This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group \({\text {GSpin}}(n,2)\). We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.

Gross和Zagier推测，在正积分谱参数s下，椭圆模群的自守格林函数\（G_s（z_1，z_2）\）的CM值（某些Hecke平移）由合适类别中的代数对数给出领域。我们证明了这个猜想的局部平均形式，我们将固定判别式\（d_1 \）的所有CM点上的第一个变量\（z_1 \）相加（由属字符扭曲），并在第二个变量中允许判别\（d_2 \）在各个CM点的评估。该结果是从与该群体相关的Shimura品种的自守格林函数的更一般的陈述中得出的\（{\ text {GSpin}}（n，2）\）。我们还使用我们的方法证明了模块化曲线上Kuga-Sato品种上较高Heegner除数的Gross-Kohnen-Zagier定理。